A ball of mass m is attached to the lower end of a light...

A ball of mass $m$ is attached to the lower end of a light vertical spring of force constant $k$. The upper end of the spring is fixed. The ball is released from rest with the spring at its normal (unstretched) length and comes to rest again after descending through a distance $x$. (A) $x=m g / k$ (B) $x=2 m g / k$ (C) The ball will have no acceleration at the position where it was descending through $x / 2$. (D) The ball will have an upward acceleration equal to $g$ at its lowermost position.

A ball of mass $m$ is attached to the lower end of a light vertical spring of force constant $k$. The upper end of the spring is fixed. The ball is released from rest with the spring at its normal (unstretched) length and comes to rest again after descending through a distance $x$.
(A) $x=m g / k$
(B) $x=2 m g / k$
(C) The ball will have no acceleration at the position where it was descending through $x / 2$.
(D) The ball will have an upward acceleration equal to $g$ at its lowermost position.

Key Concepts

Conservation of Energy in Oscillatory Systems

When analyzing the motion of a mass attached to a spring, energy conservation is often used to understand how gravitational potential energy converts into spring potential energy and vice versa. For instance, if a mass is released from rest from the unstretched position, the condition where the gravitational work equals the elastic potential energy can determine the maximum extension reached by the mass during its motion.

Dynamics of Mass-Spring Oscillations

The motion of a mass attached to a spring involves both the instantaneous forces acting on the mass and the resulting acceleration. Even if the net force is zero at a certain displacement (as calculated through force balance), the mass may have non-zero velocity. Conversely, at the extreme positions where the velocity is momentarily zero, the net force (and thus acceleration) may be quite large, driving the subsequent motion.

Hooke’s Law

This law states that the force exerted by a spring is proportional to its extension or compression from the natural length, typically expressed as F = -kx. It is fundamental in analyzing the forces present in spring?mass systems and is used to calculate the restoring force that opposes displacement.

Static Equilibrium in Vertical Spring Systems

In a vertical spring-mass system, static equilibrium is achieved when the upward restoring force of the spring balances the downward gravitational force. This condition is given by kx_eq = mg, where x_eq is the extension at equilibrium. Although it determines the position where net force is zero, it does not necessarily represent the maximum displacement when dynamic energy exchange is considered.

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